Abstract
We examine some differential geometric approaches to finding approximate solutions to the continuous time nonlinear filtering problem. Our primary focus is a new projection method for the optimal filter infinite dimensional Stochastic Partial Differential Equation (SPDE), based on the direct L2 metric and on a family of normal mixtures. We compare this method to earlier projection methods based on the Hellinger distance/Fisher metric and exponential families, and we compare the L2 mixture projection filter with a particle method with the same number of parameters, using the Levy metric. We prove that for a simple choice of the mixture manifold the L2 mixture projection filter coincides with a Galerkin method, whereas for more general mixture manifolds the equivalence does not hold and the L2 mixture filter is more general. We study particular systems that may illustrate the advantages of this new filter over other algorithms when comparing outputs with the optimal filter. We finally consider a specific software design that is suited for a numerically efficient implementation of this filter and provide numerical examples.
Highlights
In the nonlinear filtering problem, one observes a system whose state is known to follow a given stochastic differential equation
When the observations are made in continuous time, the probability density follows a stochastic partial differential equation known as the Kushner–Stratonovich equation
We show that the projection filter for basic mixture manifolds in L2 metric is equivalent to a Galerkin method
Summary
In the nonlinear filtering problem, one observes a system whose state is known to follow a given stochastic differential equation. We will write down the stochastic ODE determined by the geometric approach when H = L2 and show how it leads to a numerical scheme for finding approximate solutions to the Kushner–Stratonovich equations in terms of a mixture of normal distributions. We will call this scheme the L2 normal mixture projection filter or the L2NM projection filter. 3, we introduce the geometric structure we need to project the filtering SPDE onto a finite-dimensional manifold of probability densities.
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